A (binary) relation$\sim$ on a set $A$ is total if any two elements are related in one order or the other:

$\forall (x, y: A),\; x \sim y \;\vee\; y \sim x$

In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is total if its union with its reverse is the universal relation:

$A \times A \subseteq R \cup R^{op}$

Of course, this containment is in fact an equality.

Note that an entire relation is sometimes called ‘total’, but these are unrelated concepts. The ‘total’ there is in the sense of a total (as opposed to partial) function, while the ‘total’ here is in the sense of total (as opposed to partial) order.

Last revised on December 16, 2016 at 07:13:45.
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